Lemma 19.7.25E41D68AC6F7C

Proof. We first observe that the cone $C(S)\cap C(S')$ has full dimension since otherwise $\operatorname{vol}(S\cap S')=0$. Take a full-dimensional subcone $C'$ in $C(S)\cap C(S')$ such that $C'$ intersects the boundary of $C(S)\cap C(S')$ only at $0$. It follows from [Theorem 1, KK12] that there is an integer $N>0$ such that any $x\in \mathbb {Z}^{n+1}\cap C'$ with Euclidean norm at least $N$ lies in $S\cap S'$. Therefore, $S\cap S'\in \mathcal{S}'(C)$. Since $C'$ can be chosen around any ray in the interior of $C(S)\cap C(S')$, the same argument also shows that

This completes the proof. $\square$